Convergent sequence

**Rozaline** · October 25th 2009, 03:30 PM

I am having a lot of trouble attempting this problem. I do not know what is the best way to go about proving it.

Suppose sequence $a_n > 0$ and $b_n = a_n + \frac{1}{a_n}<br />$
Assume $a_n$ >= 1 for all n, and that $b_n$ converges. Prove that $a_n$ converges.

If it assumed that $b_n$ converges but only $a_n > 0$ , it does not follow that $a_n$ converges. Find the example.

**Opalg** · October 26th 2009, 01:18 AM

Originally Posted by Rozaline

I am having a lot of trouble attempting this problem. I do not know what is the best way to go about proving it.

Suppose sequence $a_n > 0$ and $b_n = a_n + \frac{1}{a_n}<br />$
Assume $a_n$ >= 1 for all n, and that $b_n$ converges. Prove that $a_n$ converges.

If it assumed that $b_n$ converges but only $a_n > 0$ , it does not follow that $a_n$ converges. Find the example.

If $b_n = a_n + \tfrac{1}{a_n}$ then $a_n^2 - b_na_n + 1 = 0$ . Use the quadratic formula to solve that equation: $a_n = \tfrac12\bigl(b_n\pm\sqrt{b_n^2-4}\bigr)$ . That should tell you enough about $a_n$ to find solutions for both parts of the problem.

Thread: Convergent sequence

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